Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.4.3.20. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories, and let $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be the fiber of $q$ over an object $D \in \operatorname{\mathcal{D}}$.Then the canonical map $(\operatorname{\mathcal{C}}_{D})^{\simeq } \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}^{\simeq }} \operatorname{\mathcal{C}}^{\simeq }$ is an isomorphism. In other words, the inclusion functor $\operatorname{\mathcal{C}}_{D} \hookrightarrow \operatorname{\mathcal{C}}$ is conservative.

Proof. Apply Corollary 4.4.3.19 in the special case $\operatorname{\mathcal{D}}' = \{ D\} $. $\square$